Shares of tobacco companies tumbled after Philip Morris International Inc. said cigarette shipments fell more than expected and sales for its cigarette alternative started to stall in a key market—raising wider alarm about the health of the entire tobacco industry.

Shares in the New York-listed, Switzerland-operated tobacco giant ended down 16%. That was its biggest one-day slide since becoming a public company in March 2008. The disappointing first-quarter results dragged down shares of other giants, like Altria Group Inc.,...